Tonometers are frequency standards, and usually occur in sets. Rudolph
Koenig made two types: resonant bars and tuning forks.

The Tonometer at the right consists of a series
of steel bars, suspended from strings at a distance of 22.4% of the overall
length from the ends. Under these circumstances, the rods vibrate transversely
in their fundamental modes when struck with a small hammer. For a set of round
rods of constant diameter, the frequency is proportional to the inverse square
of the length. The 1889 Koenig catalogue lists the apparatus at 80 francs.
The longest bars ought to be audible to everyone, but the shortest has a
frequency of 32,768 Hz, inaudible to everyone. The hammer was extra.

The identical tonometer at the left is at Case Western Reserve
University in Cleveland, Ohio.

The bars are labeled Ut-7, Mi-7, Sol-7, Ut-8, Mi-8, Sol-8, Ut-9,
Mi-9, Sol-9 and Ut-10. The frequencies are double vibrations (twice
the conventional frequency) and are 8192, 10240, 12288, 16384, 20480, 24576,
32768, 40960, 49152 and 65536. The non-audible frequencies must have been
obtained by calculation as the bars were filed to length,

The tonometers below are at the University of Toronto, and
were almost certainly part of the purchase of part of the apparatus that
Koenig brought to the 1876 Centennial Exposition in Philadelphia. Note the
closeup of marking on the highest frequency bar, indicating that it oscillated
at 65536 vibrations per second, corresponding to a frequency of 32768 Hz.
It is interesting to note that at the age of forty one, Koenig's personal
upper limit of hearing was 23,00 Hz, declining to 20,480 at fifty seven and
18,432 Hz at the age of sixty seven. The lower tonometer sold for 150 francs
in the 1889 Koenig catalogue.

Another set of tonometer bars from the first quarter of the
twentieth century may be found elsewhere.

In February 2000 I visited the United States
Military Academy at West Point New York. Part of the apparatus that Koenig
brought to Philadelphia for the 1876 Centennial Exposition had been bought
by the United States government and sent to West Point, but I knew that the
apparatus was now in Washington, D.C. at the National Museum of American History
at the Smithsonian Institution.

Therefore, I was quite surprised to see the seven tuning
forks mounted on resonators
in the pictures at the right. The set of three carried markings, "#50 - 904",
"#51 - 912" and "#52 - 920", and the set of four were marked "#58 - 968",
"#59 - 976", "#60 - 984" and "#61 - 992". These were clearly part of Koenig's
tonometer, with the forks at 4 Hz intervals. (Remember that the oscillation
numbers must be divided by two to get the frequency in Hertz.)